Abstract
We prove an equivalence, called $\sigma$-deformation, between a certain class of linear differential equations (called $\sigma$-compatible) and a class of modules with a semi-linear action of an automorphism $\sigma$ (called stratified). We obtain this result under the assumption that the automorphism $\sigma$ satisfies some conditions called infinitesimality and non degeneracy. We give an application to the so called theory of finite difference equations: we generalize the results of Y.André and L.Di Vizio [ADV04] and of the author [Pul08] to the automorphisms of the form $f(T)\mapsto f(qT+h)$. We investigate then the functional equation of the Morita's $p$-adic Gamma function $\Gamma_p(T)$, that is interpreted as a finite difference equation. We prove that $\Gamma_p(T)$ is the solution of a differential equation having coefficients in the unit disk $\mathrm{D}^-(0,1)$. We then relate the radius of convergence of this particular equation to the absolute value of the coefficients of $\log(\Gamma_p(T))$, that are known to be the values at positive integers of some Kubota-Leopoldt's $p$-adic $L$-functions. We finally deduce from the above theory a family of congruences between these values, we find expressions similar to that of L.Washington [Was98] and D.Barsky [Bar83]. The first part of the paper is devoted to introduce and develop the notion of \emph{radius of convergence function} on the Berkovich space of an 1-dimensional affinoid, following F.Baldassarri and L.Di Vizio [BV07]. We give an alternative proof of the continuity of the radius of convergence function in this context.